#6353045

order	512
nilpotency class	3
isoclinism	isoclinic class of #6352941
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2730 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2,7,5)(3,11,14,16)(4,10,6,9)(8,15,13,12)(18,22)(20,27)(23,31)(24,28)(25,29)(30,32);(1,3)(2,8)(4,12)(5,13)(6,15)(7,14)(9,11)(10,16)(17,18)(19,23)(20,28)(21,24)(22,27)(25,30)(26,32)(29,31);(1,4,7,6)(2,9,5,10)(3,12,14,15)(8,11,13,16)(17,19,21,26)(18,23,24,32)(20,29,27,25)(22,30,28,31);(1,5,7,2)(3,13,14,8)(4,10,6,9)(11,12,16,15);(1,4,7,6)(2,10,5,9)(3,12,14,15)(8,16,13,11)

Polycyclic presentation:

2^9|a^2=hi,c^2=h,d^2=hi,e^2=hi,f^2=h,b^a=bf,c^a=cg,f^a=fh,f^b=fh,g^b=gi,f^c=fi,e^d=ehi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6260241);